Is there a relatively simple algorithm for factoring polynomials in $GF(2^k)[x]$ with $32 \leq k \leq 128$?
I like McEliece's algorithm for factoring polynomials in $GF(2)[x]$, because of its simplicity, but the runtime is at least proportional to the cardinality of the coefficient field, so it would be very slow in $GF(2^k)[x]$ for large k.
The polynomials I am working with are all reducible to linear terms $(x-a)(x-b)(x-c)\ldots$.